Origin and usage of “safe and sound”

It is the ancient and still used meaning of sound of free from injury, healthy that is used in the common saying "safe and sound":

Sound:

• "free from special defect or injury," c.1200, from Old English gesund "sound, safe, having the organs and faculties complete and in perfect action," from Proto-Germanic *sunda-, from Germanic root *swen-to- "healthy, strong".

Safe and sound:

• Out of danger and unharmed, as in It was a challenging climb, so I'm relieved they got home safe and sound. [c. 1300 ]

(AHD)

Early usage examples:

1594, William Shakespeare, The Comedy of Errors, act 4, scene 4,:

• Fetch our stuff from thence: I long that we were safe and sound aboard.

1570 The Scholemaster: Or Plaine and Perfite Way of Teachyng Children:

• Who, by his wisedome and honestie, by his example and authority, may be able to kepe them safe and sound, in the feare of God...

Geoffrey Chaucer (1340-1400) uses hole and sounde with the meaning of safe and sound.

OED - s.v. sound a.] sound, healthy; safe, unharmed [See hole3 adj., sounde2 adv.]

(A Lexical Concordance to the Works of Geoffrey Chaucer)

If you look in the dictionary, you'll find two definitions for the word 'sound':

Sound (2): "free from injury, damage, defect, disease, etc.; in good condition; healthy; robust"

As in 'sound in body and mind' which is used in a legal context for being healthy mentally and physically.

So 'sound' does really mean 'whole' it's just not that well-used. And words phrases stay in use arguably forever once they become idiomatic. You can see a few here that clearly have very little relevance now but are still widely understood today.

This explanation from here tells it all:

Meaning 'unharmed, free from injury,' this phrase was first recorded in Middle English around the beginning of the 14th century C.E. It dates to a time when the secondary definition of sound - 'whole, not damaged or injured' - was more commonly used. (It is this definition that gives us the expression of sound body and mind, which is still used today.) With this older meaning in mind, the phrase is actually very straightforward. In ancient or modern times, it was always a good thing to return to your loved ones with all of your body parts still attached and functional.

Sound is still used in the sense conveyed in this passage, but it is largely limited to this phrase. However, this definition remains on MW:

free from flaw, defect, or decay

If you want to go further in depth, this Wiktionary article describes the transition process much more:

From Middle English sound, sund, isund, ȝesund, from Old English sund, ġesund (“sound, safe, whole, uninjured, healthy, prosperous”), from Proto-Germanic *gasundaz, *sundaz (“healthy”), from Proto-Indo-European *sunt-, *swent- (“vigorous, active, healthy”). Cognate with Scots sound, soun (“healthy, sound”), Saterland Frisian suund, gesuund (“healthy”), West Frisian sûn (“healthy”), Dutch gezond (“healthy, sound”), Low German sund, gesund (“healthy”), German gesund (“healthy, sound”), Danish sund (“healthy”), Swedish sund (“sound, healthy”), Irish fétaid (“to be able”). Related also to Dutch gezwind (“fast, quick”), German geschwind (“fast, quick”), Old English swīþ (“strong, mighty, powerful, active, severe, violent”). See swith.

The following Ngram also shows that usage really jumped in the eighteenth century (if you're interested in prevalance):

Don't fear: you can still clearly use the archaic form of sound and be understood.

Hope I could help.